In particular, for the approach based on binary weights in the FC layer “binary, FC w.” we have considered the binarization of the sole weight matrix \({W}\in \mathbb {R}^{d_{\text{out}} \times d_{\text{in}}}\)  (the bias vector of the FC layer is therefore represented as single precision floating point number). Moreover, as binarization function we have used the \(\mathrm {sign}(\cdot )\)  function. Precisely, for the backward pass of the \(\mathrm {sign}(\cdot )\)  function, the straight-through estimator of the gradient is used, which is defined as \(g_{r} = g_{q} 1_{\vert r\vert \le 1}\) , being \(r\)  and \(q\) , the input and output of the \(\mathrm {sign}(\cdot )\)  function, respectively (see [1]}). Furthermore, as recommended in [2]}, we scale the binary weights by \(\alpha =\Vert {W} \Vert _1/(d_{\text{out}} \cdot d_{\text{in}})\) . Thus, we can write
\(x_{\text{out}} = \alpha W_b x_{\text{in}} + b\) ,
where \(x_{\text{in}}, W_b, b, x_{\text{out}}\)  denote the input of the binary FC layer, the binary weight matrix, the bias vector, and the output of the binary FC layer, respectively. With this approach, we could reduce the number of parameters in terms of bits in the encoder layer by about \(75 \%\) . Another approach is based on the network pruning and fine tuning, “pruning + FT”. More in details, after having chosen the best configuration as described in Section REF , we set to zero all the weights with the smallest absolute value, according to a predetermined percentage, e.g., \(40\%\) . As expected, this degrades the performance. Therefore, after pruning the network, we fine tune the network by optimizing the remaining parameters until the stopping criterion is met. In particular, as highlighted in Tables REF  and REF , the method “pruning + FT” helps to maintain approximately the same performance as the method without pruning and at the same time reduces the number of parameters to be offloaded to the MT by \(40\%\) . On the other hand, the approach “binary FC w.” produces a higher degradation of the performances. Finally, we have compared our method with other two approaches based on AE NNs. In particular, we see that our method outperforms both, the CsiNet [3]} and the recent ACRNet presented in [4]}. Differently from our approach, CsiNet and ACRNet have been trained on DL CSI data, and the data have been normalized as specified in the respective works. The two networks leverage the sparsity of the CSI in the delay domain. However, ACRNet boosts the performance by making use of network aggregation in the decoder and by replacing the rectified linear unit (ReLU) activation with a parametric ReLU (PReLU) activation.
